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Topic: Rank of a matrix

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	Rank (linear algebra) - Wikipedia, the free encyclopedia
		In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent.
		The rank of a matrix plus the nullity of the matrix equals the number of columns of the matrix (this is the "rank theorem" or the "rank-nullity theorem").
		Numerical determination of rank requires a criterion for deciding when a value, such as a singular value from the SVD, should be treated as zero, a practical choice which depends on both the matrix and the application.
	en.wikipedia.org /wiki/Rank_of_a_matrix (730 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-06)
		The dimension of this space is called the "row rank" of the matrix.
		There are theorems in linear algebra that prove that the row rank of a matrix always equals the column rank, and the common value is just called the "rank" of the matrix.
		The number of non-zero rows in the row-echelon form is the row rank of the matrix, and therefore the rank.
	mathforum.org /library/drmath/view/52010.html (351 words)

	Matrices
		Rank: the rank of a matrix A, often denoted r(A), is the number of linearly independent rows or columns of that matrix.
		It is easily shown that the row rank of a matrix is equal to the column rank of a matrix.
		Trace: the trace of a matrix A (denoted trA) is the sum of the elements on the principal diagonal, i.e.
	cepa.newschool.edu /het/essays/math/matrix.htm (945 words)

	æstats - r a n k matrix
		note: the matrix used to calc the stats on the dm rank page is 200 x 200 in size.
		This is a subset of the full ranking matrix, since it is incident sorted, the folks that played the most are found in the top left.
		This matrix shows the same data as the one above, only now the data is presented as an intensity range.
	www.hostultra.com /~clanstnt/stats/AEstatsRankMatrix.html (182 words)

	Matrix Algebra - Review
		A matrix is a rectangular array of elements arranged in rows and columns.
		The rank of a matrix is defined as the maximum number of linearly independent columns in the matrix.
		Note that the inverse of a square matrix of dimension r x r exists iff the rank of the matrix is r.
	condor.depaul.edu /~jmorgan1/csc334.matalg.html (905 words)

	Matrices: Rank (Site not responding. Last check: 2007-11-06)
		of a matrix is a very important concept in the use of matrices to solve a set of linear equations.
		is also used to identify the linear independence of the rows (or columns) of a matrix.
		of the transpose of a matrix is the same as the original matrix
	www.rit.edu /~pnveme/pigf/Matrices/mat_rank_2.html (79 words)

	OnLine2-3-RandMats.html
		Since the rank of a matrix is the number of nonzero rows in the reduced echelon form of the matrix, it is bounded by the minimum of m and n.
		It is worthwhile to note that the rank of a random matrix is usually actually equal to the minimum of m and n.
		In both cases this corresponds to the number of pivots in reduced echelon form of the matrix, that is, the rank of the matrix.
	adept.maplesoft.com /powertools/linearalgebra/html/OnLine2-3-RandMats.html (820 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Example of a square matrix: B = EMBED Equation.3 Vector A special case of a matrix is a vector (or a column vector).
		Identity Matrix, Denoted I, is a diagonal matrix with 1's on the diagonal.
		Rank of a Matrix The rank of a matrix A, r(A), is the maximum number of linearly independent columns (vectors) or rows (transposes of vectors) of the matrix.
	statweb.calpoly.edu /jdaly/stat324/matrices.doc (1885 words)

	Question 4 (Site not responding. Last check: 2007-11-06)
		The coefficient matrix is not rank zero, since it is not zero.
		It is not rank one, since otherwise all the left-hand sides of the equations would be proportional.
		The coefficient matrix is not rank one, since the left-hand sides of the equations are not proportional and is clearly not rank zero either.
	www.math.pitt.edu /~sparling/118033/118033e1s/node5.html (325 words)

	Journal of the American Statistical Association: On the asymptotic properties of LDU-based tests of the rank of a ... (Site not responding. Last check: 2007-11-06)
		On the asymptotic properties of LDU-based tests of the rank of a matrix.
		Journal of the American Statistical Association; 9/1/1996; Donald, Stephen G. The LDU decomposition-based test for the rank of a matrix proposed by L. Gill and A. Lewbel is not widely applicable as it is believed to be because its critical values are difficult to compute even asymptotically.
		Gill and Lewbel recently introduced a test for the rank of a matrix based on the LDU decomposition.
	www.highbeam.com /library/doc0.asp?DOCID=1G1:18972963&refid=ip_encyclopedia_hf (211 words)

	æstats - d i f f matrix (Site not responding. Last check: 2007-11-06)
		The so called difference matrix shows rank matrix data in a slightly different way, by subtracting a player's deaths from his frags.
		note: the matrix used to calc the stats on the dm rank page is 4 x 4 in size.
		This is the subset list of of all the players in ranking, the folks with the most incidents (20 of the total 4).
	www.redrival.com /cylune/AEstatsDiffMatrix.html (182 words)

	Rank of a Matrix (Site not responding. Last check: 2007-11-06)
		The matrix has a zero determinant and is therefor singular.
		If you look the matrix you see that it has two identical rows (and two identical columns).
		The rank of a matrix is the maximum number of independent rows (or, the maximum number of independent columns).
	chortle.ccsu.ctstateu.edu /VectorLessons/vmch16/vmch16_15.html (133 words)

	Rank (matrix theory) - InfoSearchPoint.com (Site not responding. Last check: 2007-11-06)
		The column rank and the row rank are indeed equal and is simply called the rank of A.
		It is commonly denoted by either rk(A) or rank A.
		There are different generalisations of the concept of rank to matrices over arbitrary ring.
	www.infosearchpoint.com /display/Rank_(matrix_theory) (704 words)

	Mailgate: sci.stat.math: Re: rank deficiency of covariance matrix (Site not responding. Last check: 2007-11-06)
		Now if the mean of the vectors is subtracted from each one and the n X n covariance matrix C = XX^T is formed, C will have the same rank as X. (Note that because of the mean subtraction, M >= n+1 for full rank).
		Theoretically, the rank, k, is the number of nonzero eigenvalues of C. The fun starts when you have to determine which of the small eigenvalues should be considered zero when truncation and roundoff error are taken into account.
		The fun continues when you have to determine which of the remaining small eigenvalues should be discarded because the fraction of spread information contained in the corresponding PCA directions is negligible.
	mailgate.supereva.it /sci/sci.stat.math/msg09989.html (354 words)

	Introduction to Econometrics - Matrix Algebra (Site not responding. Last check: 2007-11-06)
		A special diagonal matrix is a matrix where all diagonal elements are equal to one (identity matrix denoted I).
		The rank of a (mn) matrix* is equal to the rank of the largest sub matrix with a determinant different from zero where the determinant of a matrix is defined by
		The determinant of an orthogonal matrix is equal to 1 or -1.
	www.resacorp.com /matrix_algebra.htm (1824 words)

	Let's make all this useful
		Theorem: The rank of a matrix A equals the maximum number of linearly independent column vectors.
		Consequently, a matrix has the same number of linear independent row vectors as it has linear independent column vectors.
		Two non-zero rows (row vectors) remain, therefore the rank of the matrix is 2.
	www.math.fsu.edu /~fusaro/EngMath/Ch6/PML.html (1004 words)

	M244 Linear Lab on Determinants and Inverses (Site not responding. Last check: 2007-11-06)
		A method for finding the inverse of a square matrix (if it exists) is to adjoin the nxn identity matrix to it and then bring the adjoined matrix into reduced echelon form.
		The rank of a matrix is the number of nonzero rows in the row-reduced echelon form for the matrix.
		Conjectured Property: The determinant of a matrix with two equal rows or columns is _____________________________.
	www.saintjoe.edu /~karend/m244/NBK3-1.html (733 words)

	3.1. Definitions and terminology
		Proposition 1 The rank of a matrix A is the number of
		the rank of A + the nullity of A = n.
		The elementary row operations do not affect the rank of a matrix A.
	www.ee.oulu.fi /~mpa/matreng/ematr3_1.htm (223 words)

	Strongly Consistent Determination of the Rank of Matrix ewp-em/0307007 (Site not responding. Last check: 2007-11-06)
		Using the matrix perturbation theory to construct or find a suitable bases of the kernel (null space) of the matrix and to determine the limiting distribution of the estimator of the smallest singular values.
		The test, based on matrix perturbation theory, enable to determine how many singular values of the estimated matrix are insignificantly different from zero and we fully characterise the asymptotic distribution of the generalized Wald statistic under the most general conditions.
		We establish a strongly consistent of the determination of the rank of matrix using the two approaches.
	econwpa.wustl.edu /eprints/em/papers/0307/0307007.abs (382 words)

	æstats - r a n k matrix (Site not responding. Last check: 2007-11-06)
		note: the matrix used to calc the stats on the dm rank page is 125 x 125 in size.
		This page only shows a top left subset (60 x 60) of the rank matrix, that is incident sorted.
		This is the subset list of of all the players in ranking, the folks with the most incidents (60 of the total 125).
	arago4.tn.utwente.nl /unreal/Stats/AEstatsRankMatrix.html (139 words)

	[No title]
		Input, integer RANK, the "rank" of the object.
		Input, integer RANK, the "rank" of the data in the packed array A. The rank is the number of indices used to access any particular entry.
		A scalar has rank 0, a vector rank 1, a matrix rank 2, and so on.
	orion.math.iastate.edu /burkardt/f_src/packer/packer.f90 (967 words)

	[No title]
		Note that calculation of the rank of a matrix is * often numerically difficult, especially in the case of a near singular * matrix.
		While GEN1 is used to extract an * element from a vector such as M, the MATRIX command must be used to extract elements * from a matrix.
		The general format is: * * COPY fromvar(s) tovar / options * * where fromvar(s) is either a list of vectors or a single matrix, tovar is * the variable into which the fromvar(s) are to be copied, and options is a * list of desired options.
	www.ucs.mun.ca /~noelroy/6002/Greene/AppendixA.sha (1055 words)

	[No title]
		A matrix with one row is called a row vector, and a matrix with one column is called a column vector.
		e.g.: ó¨Matrix scalar multiplication¨L Multiplication of a matrix or a vector by a scalar is also straightforward:ó¨Transpose of a matrix¨0 Taking the transpose of a matrix is similar to that of a vector: The diagonal elements in the matrix are unaffected, but the other elements are switched.
		To find the rank of a matrix by hand, use Gauss elimination and the linearly dependant row vectors will fall out, leaving only the linearly independent vectors, the number of which is the rank.
	www.eng.fsu.edu /~palanki/orient/lec1.ppt (1262 words)

	CPACT Software Home - PreScreen
		Selecting Rank Correlation Matrix from the top menu will calculate this matrix, then visualise it as a palette of colours.
		A vertical colour scale is appended on the right hand side of the axes of the matrix.
		Mouse pointing on the matrix plot displays the absolute value of the rank correlation between the two pointed variables inside the Information frame.
	www.ncl.ac.uk /CPACTsoftware/PreScreen/RankCorr.html (95 words)

	R Help 2002: Re: [R] function for rank of a matrix ? (Site not responding. Last check: 2007-11-06)
		Re: [R] function for rank of a matrix ?
		In reply to: Raphael Gottardo: "[R] function for rank of a matrix ?"
		rank numerically is an ill-defined operation due to floating-point
	www.r-project.org /nocvs/mail/r-help/2002/0305.html (290 words)

	Math 307 §E1: Objectives for Second Midterm Exam (Site not responding. Last check: 2007-11-06)
		Compute the determinant of a matrix by cofactor expansion, exploiting row and column operations to simplify.
		Represent elements of the inverse of a matrix in terms of cofactors.
		Relate the rank of a matrix to other properties in the invertible matrix theorem.
	orion.math.iastate.edu /alex/307/mt2_obj.html (205 words)

	Wilmott Forums - Rank of a Matrix
		In the same way standardize each row of the matrix, so that the entire first column is ‘1’.
		This is taking the matrix to a form where the first element different from zero in each non-zero row is equal to 1, and any other element in the same column but lower row is zero.
		This is actually the way of finding the rows of the matrix that are LI without ambiguity.
	www.wilmott.com /messageview.cfm?catid=4&threadid=2729 (395 words)

	SIMAX Volume 10 Issue 4
		Gill, Golub, Murray, and Saunders have described five methods by which the Cholesky factors of a positive definite matrix may be updated when the matrix is subjected to a symmetric rank 1 modification.
		For a negative rank 1 update, a modification of one of their methods was given by Lawson and Hanson and analyzed by Bojanczyk, Brent, van Dooren, and de Hoog.
		In their comparison, the authors do not consider pipelining two applications of the rank 1 algorithms, which in certain instances is possible.
	locus.siam.org /SIMAX/volume-10/art_0610041.html (222 words)

	[No title]
		A single-output system is observable iff in the Jordan form matrix there is one Jordan block associated with each distinct eigenvalue and every entry of C corresponding to the first column of each Jordan block is nonzero.
		Determine the matrix Q for the transformation: If rank(A) is n1, the first n1 columns of Q are any n1 linearly independent columns from the controllability matrix.
		Determine the matrix P for the transformation: If rank(A) is n2, the first n2 rows of P are any n2 linearly independent rows from the observability matrix.
	www.csus.edu /indiv/n/ngw/EEE-241/con-ob.doc (1874 words)

	Mailgate: sci.stat.math: Re: rank deficiency of covariance matrix (Site not responding. Last check: 2007-11-06)
		Mohammed Ali wrote: > am getting a warning of "rank deficiency" of covariance matrix when I > apply principal component analysis to reduce dimension of feature vector > for a classification problem.
		Having a rank deficient matrix is not by itself a problem, PCA works just fine on rank deficient matrices as well as on full-rank matrices.
		As the post by Greg Heath has explained, and I agree with him, PCA may or may not be a useful pre-processing tool for classification.
	mailgate.supereva.it /sci/sci.stat.math/msg09992.html (137 words)

	Science Forums and Debate - Rank of a n x m matrix
		In the process, I am in the need of knowing an algorithm to determine the rank of a matrix.
		I am dealing with a 4x3 matrix and need a method (that can be easily coded into a computer programme) which would mechanically compute the rank of such a matrix.
		I must add that determination of rank using the definition of rank seems a bit too inefficient to code.
	www.scienceforums.net /forums/printthread.php?t=5514 (219 words)

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